An indirect method for computing origins for Hopf bifurcation in two-parameter problems

Branches of turning points and of Hopf bifurcation points can exist in a two-parameter problem because simple turning points and Hopf points are structurally stable in the one-parameter problem. It is well-known that in a two-parameter problem a branch of Hopf points can emanate from a turning point curve at a so-called BT-point for which the Jacobian matrix of the steady state problem has a double eigenvalue zero with a one-dimensional nullspace. Hence such a BT-point that is also called origin for Hopf bifurcation can be detected during the continuation of a turning point curve. In order to determine such a BT-point efficiently we present an indirect method that fits to the continuation procedure. Based on a nonsingular parametrization of the turning point curve a scalar test function is given, for which the parameter value of the BT-point is a regular zero. A Newton-like method that utilizes the internal structure of the test function and it's derivative is used to generate a superlinearly convergent sequence of parameter values. At each step the corresponding turning point has to be calculated approximately by a direct method. The implementation of the whole procedure is organized in such a way that the second order derivatives of the original problem occur only in form of directional derivatives and, hence, they are approximated by special difference formulas requiring only a few additional function values. Numerical examples illustrate the behavior of the algorithm, where one example is a model of a chemical exothermic reaction described by a system of partial differential equations.